Exponential-Size Circuit Complexity is Comeager in Symmetric Exponential Time

2026-05-04

AI Summary

This work investigates whether languages requiring exponential-size circuits are prevalent within the symmetric exponential-time class $S^E_2$. By extending resource-bounded category theory to $S^E_2$ for the first time, the authors introduce a notion of "meagerness" via a Banach–Mazur game defined through single-valued $FS^P_2$ strategies. Leveraging an $FS^P_2$ algorithm for the Range Avoidance problem, they construct a winning strategy that establishes the meagerness of SIZE($2^n/n$) in $S^E_2$. Consequently, languages necessitating exponential circuit size form a comeager set in $S^E_2$, demonstrating that exponential circuit complexity is indeed typical within this class.

Abstract

Lutz (1987) introduced resource-bounded category and showed the circuit size class SIZE($\frac{2^n}{n}$) is meager within ESPACE. Li (2024) established that the symmetric alternation class $S^E_2$ contains problems requiring circuits of size $\frac{2^n}{n}$. In this note, we extend resource-bounded category to $S^E_2$ by defining meagerness relative to single-valued $FS^P_2$ strategies in the Banach-Mazur game. We show that Li's $FS^P_2$ algorithm for the Range Avoidance problem yields a winning strategy, proving that SIZE($\frac{2^n}{n}$) is meager in $S^E_2$. Consequently, languages requiring exponential-size circuits are comeager in $S^E_2$: they are typical with respect to resource-bounded category.

Problem

• circuit complexity
• resource-bounded category
• symmetric alternation
• exponential-size circuits
• comeager

Innovation

• resource-bounded category
• circuit complexity
• symmetric alternation
• Banach-Mazur game
• comeager